A071675 Array read by antidiagonals of trinomial coefficients.

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 0, 3, 3, 1, 0, 0, 2, 6, 4, 1, 0, 0, 1, 7, 10, 5, 1, 0, 0, 0, 6, 16, 15, 6, 1, 0, 0, 0, 3, 19, 30, 21, 7, 1, 0, 0, 0, 1, 16, 45, 50, 28, 8, 1, 0, 0, 0, 0, 10, 51, 90, 77, 36, 9, 1, 0, 0, 0, 0, 4, 45, 126, 161, 112, 45, 10, 1, 0, 0, 0, 0, 1, 30, 141, 266, 266

Offset: 0

Author

Henry Bottomley, May 30 2002

Keywords

• nonn
• tabl

Comments

Read as a number triangle, this is the Riordan array (1, x(1+x+x^2)) with T(n,k) = Sum_{i=0..floor((n+k)/2)} C(k,2i+2k-n)*C(2i+2k-n,i). Rows start {1}, {0,1}, {0,1,1}, {0,1,2,1}, {0,0,3,3,1},... Row sums are then the trinomial numbers A000073(n+2). Diagonal sums are A013979. - Paul Barry, Feb 15 2005

Let {a_(k,i)}, k>=1, i=0,...,k, be the k-th antidiagonal of the array. Then s_k(n)=sum{i=0,...,k}a_(k,i)* binomial(n,k) is the n-th element of the k-th column of A213742. For example, s_1(n)=binomial(n,1)=n is the first column of A213742 for n>1, s_2(n)=binomial(n,1)+binomial(n,2)is the second column of A213742 for n>1, etc. In particular (see comment in A213742) in cases k=4,5,6,7,8, s_k(n) is A005718(n+2), A005719(n), A005720(n), A001919(n), A064055(n+3), respectively. - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012

Examples

			Rows start
1, 0,  0,  0,  0,  0, ...;
1, 1,  1,  0,  0,  0,  0, ...;
1, 2,  3,  2,  1,  0,  0, ...;
1, 3,  6,  7,  6,  3,  1, 0, ...;
1, 4, 10, 16, 19, 16, 10, 4, 1, ...; etc.
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
• Milan Janjic, Binomial Coefficients and Enumeration of Restricted Words, Journal of Integer Sequences, 2016, Vol 19, #16.7.3

Crossrefs

Visible version of A027907. Row sums are 3^n, i.e. A000244. Central diagonal is A002426. Cf. A071676 for a slight variation.

Programs

Mathematica

T[n_, k_] := Sum[Binomial[n - k - j, j]*Binomial[k, n - k - j], {j, 0,
Floor[(n - k)/2]}]; Table[T[n, k], {n,0,10}, {k,0,n}] // Flatten (* G. C. Greubel, Feb 28 2017 *)

Formula

T(n, k) = T(n-1, k) + T(n-1, k-1) + T(n-1, k-2) starting with T(0, 0)=1. See A027907 for more.

As a number triangle, T(n, k) = Sum_{i=0..floor((n-k)/2)} C(n-k-i, i) * C(k, n-k-i). - Paul Barry, Apr 26 2005